A question of Norton-Sullivan and the rigidity of pseudo-rotations on the two-torus

Expositor

Jian Wang

IMPA

Data

Terça-feira, 20 de novembro de 2018, 15:30

Local

Sala 228

Resumo

In 1996, A. Norton and D. Sullivan asked the following question: If $f : \mathbb{T}^2 \rightarrow \mathbb{T}^2$ is a diffeomorphism, $h : \mathbb{T}^2\rightarrow\mathbb{T}^2$ is a continuous map homotopic to the identity, and $hf = T_{\rho}h$ where $\rho \in \mathbb{R}^2$ is a totally irrational vector and $T_{\rho} : \mathbb{T}^2 \rightarrow \mathbb{T}^2 , z \mapsto z + \rho$ is a translation, are there natural geometric conditions (e.g. smoothness) on $f$ that force $h$ to be a homeomorphism? In this talk, we give a negative answer to this question with respect to the regularity. We also show that under certain boundedness condition, a $C^r$ (resp. Hölder) conservative irrational pseudo-rotation on $\mathbb{T}^2$ with a generic rotation vector is $C^{r-1}$-rigid (resp. $C^0$-rigid). These provide a partial generalization of the main results in [Bramham, Invent. Math. 199 (2), 561-580, 2015; A. Avila, B. Fayad, P. Le Calvez, D. Xu, Z. Zhang, arXiv: 1509.06906v1]. These are joint works with Zhiyuan Zhang and Hui Yang.